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Laser-induced double-dark resonances and double-transparencies in a four-level system
1 February 2002
Optics Communications 202 (2002) 97–102 www.elsevier.com/locate/optcom
Laser-induced double-dark resonances and double-transparencies in a four-level system Yong-Fang Li a,b,*, Jian-Feng Sun a, Xian-Yang Zhang a, Yong-Chang Wang b b
a College of Physics and Information Technology, Shaanxi Normal University, Xi’an 710062, People’s Republic of China Department of Applied Physics, Institute of Modern Physics, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China
Received 30 August 2001; received in revised form 12 November 2001; accepted 5 December 2001
Abstract We study the generations of dark resonance and electomagnetically induced transparency (EIT) effects in the fourlevel system with two coherent fields coupling the transition from upper level to a pair of ground levels and a lower excited state. The quantum interference between two excited pathways results in double-dark resonances and doubletransparency holes with adjustable frequencies driven into the absorption spectrum in the atomic system. A clear analytical explanation for our numerical results is presented. It is shown that the relation of detunings of two coherent fields plays an important role in the generations of the coherent population trapping and EIT effects. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 42.50.Hz; 42.65.)k
If an atom is prepared in a coherent superposition state, it is possible to cancel absorption or emission under certain condition. This interesting phenomenon is called coherent population trapping (CPT) or dark resonance; it is by now a wellknown concept in optics and laser spectroscopy. Based on this effect, there have been many applications including electromagnetically induced transparency (EIT) [1–5], lasing without inversion [6–10] the resonant enhancement of the refractive index [11–13] and adiabatic population transfer [14]. These applications have been discussed for
*
Corresponding author. E-mail address: yfl[email protected] (Y.-F. Li).
conventional use in the system with either K or N level configuration. In the V-system, Zhou Peng and Swain have performed many works on the ultrasharp lines and the role of quantum interference in probe absorption [15,16]. In the three-level K-system, however, a single dark state was generated. With the growth of experimental and theoretical studies, many new and more complex nonlinear behaviors associated with multilevel atoms interacting with multifrequency laser fields have been considered. In this paper we present a four-level system driven by two coherent fields and discuss in detail the process of the populations of each level and the population transfer between levels. We find double-dark resonances and double-EIT
0030-4018/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 2 ) 0 1 0 7 2 - 6
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Y.-F. Li et al. / Optics Communications 202 (2002) 97–102
with adjustable frequencies driven into the absorption spectrum that could not occur in the three-level system. We investigate various features of the double-dark resonances and doubleEIT. In comparison with the three-level K-system, we find that the equality of detunings of the two coherent driven fields is of importance to the generation of the double-dark resonances and double-EIT. The double-dark resonances have many interesting applications such as unipolar and bipolar quantum well lasers and the possibility of using double-dark states in adiabatic passage techniques in the quantum computation is also intriguing [17]. The model under consideration has four levels, as displayed in Fig. 1. The transitions from the upper level j3i with energy hx3 to a pair of ground levels j1i with energy hx1 and j2i with energy hx2 and the lower excited j4i with energy hx4 are driven by two coherent fields with frequencies xL1 , xL2 , respectively. These transitions are also coupled by the vacuum modes. If the separations of j3i from j1i and j2i are much larger than that from j4i or vice versa, we can consider that the vacuum mode coupling j3i to j1i and j2i is totally different from that coupling j3i to j4i. Meanwhile, we assume that the vacuum model which couples j3i to j1i and j2i is different. We also assume that the transitions between j4i and j2i, j1i are dipole forbidden. In the absence of coherent field with frequency xL2 the system becomes a three-level system where the upper transition level is coupled by a single driven field with frequency xL displayed in Fig. 1(b). In the absence of splitting of two ground
(a)
(b)
(c)
Fig. 1. Schematic diagrams of the energy levels: (a) four-level system driven by two coherent fields; (b) three-level system driven by a single coherent field; (c) three-level K-system driven by two coherent fields.
levels, the system becomes a typical K-system, as shown in Fig. 1(c). We investigate double-dark states (CPT states) and double-EIT effects corresponding to various level-systems and present a clear analytical explanation for the numerical results. The Hamiltonian in the frame rotating with the laser frequencies xL1 , xL2 is in the form H^ ¼ hD1 A11 hD2 A22 hD3 A44 þ ½X1 A31 þ X1 A32 þ X1 A34 þ H :c:;
ð1Þ
where D1;2 ¼ x3 x1;2 xL1 , D3 ¼ x3 x4 xL2 , are the detunings. Xi ði ¼ 1; 2; 3Þ is the Rabi frequency. They are defined as *
X1;2 ¼ h1 E1 l 31;2 e^1 ; X3 ¼ h
1
* E2 l 3;4
ð2Þ
e^2 ;
ð3Þ
* e 1;2
where E1;2 and are the complex amplitudes and * * basis vectors, respectively. Here, l 31;2 ¼ h3j e r j * * 1; 2i and l 34 ¼ h3j e r j4i are matrix elements of the electric dipole moments of atomic transition from j3i j1i, j2i and j3i j4i, which are assumed to be real in our system. The Alk ¼ jlihkj represents a population operator for l ¼ k and dipole transition operator for l 6¼ k. According to the generalized reservoir theory with the Weisskopf–Wigner approximation [18], and the rotating approximation, we can obtain the equations of motion for the reduced atomic density operator reported as in [19]. 1 q_ ¼ i½H ; q þ r1 ð2A13 qA31 A33 q qA33 Þ 2 1 þ r2 ð2A23 qA32 A33 q qA33 Þ 2 1 þ r3 ð2A43 qA34 A33 q qA33 Þ; ð4Þ 2 where rk ðk ¼ 1; 2; 3Þ is a spontaneous decay constant from the excited j3i to lower sublevels jki * 2 (k ¼ 1; 2; 4) and they are given by r12 ¼ jl 31;2 j ðx3 * 3 2 3 x1;2 Þ =3pe0 hc3 , r3 ¼ jl 34 j ðx3 x4 Þ =3pe0 hc3 , respectively. The equations of motion of the reduced density-matrix elements for the atomic variables take the form q_ 11 ¼ r1 q33 þ iX1 ðq13 q31 Þ; q_ 12 ¼ i½ðD1 D2 Þq12 þ X2 q13
ð5Þ X 1 q32 ;
ð6Þ
Y.-F. Li et al. / Optics Communications 202 (2002) 97–102
1 q_ 13 ¼ ðr1 þ r2 þ r3 Þq13 þ iX1 ðq11 q33 Þ 2 þ iðX2 q12 þ D1 q13 þ X3 q14 Þ;
ð7Þ
q_ 14 ¼ i½X3 q13 þ ðD1 D3 Þq14 X1 q34 ;
ð8Þ
q_ 22 ¼ r2 q33 þ iX2 ðq23 q32 Þ;
ð9Þ
1 q_ 23 ¼ ðr1 þ r2 þ r3 Þq23 2 þ iðX1 q21 þ D2 q23 þ X2 q24 Þ þ X2 ðq22 q33 Þ; ð10Þ q_ 24 ¼ i½X3 q23 ðD2 D3 Þq14 X2 q34 ;
ð11Þ
q_ 33 ¼ ðr1 þ r2 þ r3 Þq33 þ i½X1 ðq31 q13 Þ þ X2 ðq32 q23 Þ þ X3 ðq34 q43 Þ;
ð12Þ
1 q_ 34 ¼ ðr1 þ r2 þ r3 Þq34 iX3 ðq44 q33 Þ 2 iðX1 q14 þ X2 q24 Þ iD3 q34 ;
ð13Þ
q11 þ q22 þ q33 þ q44 ¼ 1:
ð14Þ
Eqs. (5)–(14) can be rewritten in the compact vector form as d ^ ^ þ I^; W ¼ L^W ð15Þ dt ^ is a column vector whose ith component where W is Wi ^5 ¼ W ^ ; W ^ 3 ¼ q13 ; W 1 ^ 13 ¼ W ^ ; W ^ 6 ¼ q22 ; W4 ¼ q14 ; W 4 ^ 10 ¼ W ^ ; W ^ 8 ¼ q24 ; W ^ 14 ¼ W ^ ; W 7 8 ^ 11 ¼ q33 ; W ^ 12 ¼ q34 ; W ^ 15 ¼ W : W ð16Þ 12
^ 1 ¼ q11 ; W ^ 9 ¼ W ; W 3 ^ 7 ¼ q23 ; W
^ 2 ¼ q12 ; W
99
j4i can be obtained by plotting the Im½q34 ð1Þ and Re½q34 ð1Þ, respectively, as a function of the variable detuning D3 . For simplicity, it is assumed that the spontaneous rates are the same for all the levels, r1 ¼ r2 ¼ r3 ¼ r. The Rabi frequencies and detunings Dij are normalized by the spontaneous rate r. First, we present the results for the case which is a three-level system with a single driving field, as shown in Fig. 1(b). Fig. 2(a) exhibits the population distribution, absorption and dispersion spectra. When the driving field is resonantly with transition either j3i j1i or j3i j2i, corresponding to D1 ¼ 0 or D1 ¼ x21 , the populations of the j1i or j2i are minimum. Only when D1 ¼ x21 =2, the population of j3i and absorption spectrum of j3i j1i ðj3i j2iÞ have a peak and the CPT state is absent. In the three-level K-system, however, we can clearly see that the population of upper level j3i is zero and the populations of the j1i and j2i are not minimum, moreover, the former has a slight increase when both driving fields are resonantly with transitions j3i j1i and j3i j2i, respectively, as shown in Fig. 2(b). For this case the populations of level j1i and j2i are about 50%, respectively. It shows that the populations are trapped in the superposition state of j1i and j2i. From the Fig. 2(b) we can see that the absorption spectrum exhibits a transparency hole. In addition, we find that as field intensities increase
L is a 15 15 matrix, and the inhomogeneous term I^ is the 15-element column vector. Hence Eq. (15) has formal solution ^ ðt þ sÞ ¼ esL^W ^ ðtÞ þ L^1 ½esL^ 1I^ W
ð17Þ
and steady-state solution is ^ ð1Þ ¼ L^1 I^: W
ð18Þ
We can numerically calculate population distribution of each level, absorption and dispersion spectra by using the steady-state solution (18). The absorption and dispersion spectra between j3i and
Fig. 2. Calculated population distribution and absorption, dispersion spectrum: (a) the three-level system with a single driving field under conditions: r1 ¼ r2 ¼ 1; X1;2 ¼ 1; x21 ¼ 2; (b) the K-system with two driving fields under conditions: r1 ¼ r2 ¼ 1; X1;2 ¼ 1; D2 ¼ 0.
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the position of the transparency hole remains unchanged, but the hole becomes wider. For the four-level system, the population feature of upper level j3i is greatly different from that of a three-level K-system; it has double-dark resonances or CPT states as shown in Fig. 3. When driving field with frequency xL2 is resonantly with transition j3i j4i and driving field with frequency xL1 is far from resonant frequency, the populations of atomic system assemble mainly at the j1i and j2i, the population of j4i is empty. But when driving field with frequency xL1 is also resonant with transition between j3i and j1i which correspond with D1 ¼ 0, the populations of j1i and j4i increase, while the population of j2i vanishes. When driving field xL1 is resonant with transition between j3i and j2i which correspond with D1 ¼ x21 , the populations of j2i and j4i increase, while the population of j1i vanishes, as shown in Fig. 3 ðq11 ; q22 ; q44 Þ. From the change of the population of each level, we can see the population transfer from the j1i; j2i to j4i in the processes of forming CPT states. In such a case, the generation of the double-dark states is similar to a double three-level K-system [20]. Fig. 4 shows that calculated population of j3i and absorption, dispersion spectral profiles are observed with the j3i j4i transition for the variable detuning of the driving field D3 . For
the case of the driving field xL2 is resonantly with the j3i j4i transition, the positions of the CPT states and transparent holes correspond just to the detuning D3 ¼ D1 or D2 , as shown in Fig. 4. The CPT states and transparence holes can be obtained with the great range of Rabi frequencies shown in Fig. 5. The Rabi frequency X3 only affects the magnitude of the absorption spectra, but does not affect the positions of the transparent holes. Based on [20] and above numerical results, we know that the processes of population trapping in our four-level system are similar to double threelevel K-system. Hence, for understanding the origination of these numerical results, we analyze the density matrix equations of the three-level Ksystem in a field-dependent basis.
Fig. 3. The population distribution for the variable detuning D1 in the four-level system under conditions: r1 ¼ r2 ¼ r3 ¼ 1; X1;2;3 ¼ 1; D3 ¼ 0; x21 ¼ 2.
Fig. 4. Calculated population and absorption, dispersion spectrum versus variable detuning D3 under the conditions: r1 ¼ r2 ¼ r3 ¼ 1; X1;2;3 ¼ 1; D1 ¼ 1; x21 ¼ 2.
j3i; jþi ¼
X1 j1i þ X2 j2i X1 j2i X2 j1i ; ji ¼ ; G G ð19Þ
where the X1;2 is the Rabi frequency of two driving fields and assuming X1;2 is real. The G is the effective Rabi ffi frequency of the pumping field qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 G ¼ X1 þ X22 . The j3i is the upper level and j1i and j2i are ground levels, as shown in Fig. 1(c). In this basis, the density matrix equations of motion can be given
Y.-F. Li et al. / Optics Communications 202 (2002) 97–102
101
Eqs. (20)–(25) can be removed by the transformations q03þ ¼ q3þ eiDt ; q03 ¼ q3 eiDt ;
q03 ¼ q3 eiDt ; q0þ3 ¼ qþ3 eiDt ;
q0 ¼ q ;
q033 ¼ q33 ;
q0þ ¼ qþ ;
q0þ ¼ qþ :
q0þþ ¼ qþþ ; ð26Þ
Substituting Eq. (26) into Eqs. (20)–(25), then the equations can be rewritten in the form Fig. 5. Calculated absorption spectra versus variable detuning D3 and Rabi frequency X3 under conditions: r1 ¼ r2 ¼ r3 ¼ 1; X1;2 ¼ 1; D1 ¼ 1; x21 ¼ 2.
i ½X1 X2 ðeiD1 t eiD2 t Þðq3 q3 Þ G ð20Þ þ ðX21 eiD1 t þ X22 eiD2 t Þðq3þ qþ3 Þ;
q_ 33 ¼ Cq33 þ
1 i q_ 3þ ¼ Cq3þ þ ½X1 X2 ðeiD1 t eiD2 t Þqþ 2 G þ q33 ðX21 eiD1 t þ X22 eiD2 t Þ i qþþ ðX21 eiD1 t þ X22 eiD2 t Þ; ð21Þ G 1 i q_ 3 ¼ r Cq3 þ X1 X2 ðeiD1 t eiD2 t Þðq q33 Þ 2 G i 2 iD1 t q3 ðX1 e þ X22 eiD2 t Þ; ð22Þ G 1 i q_ þþ ¼ Cq33 þ ðX21 eiD1 t þ X22 eiD2 t Þqþ3 2 G i 2 iD1 t ðX1 e þ X22 eiD2 t Þq3þ ; ð23Þ G i q_ þ ¼ ðX21 eiD1 t þ X22 eiD2 t Þq3 G i X1 X2 ðeiD1 t eiD2 t Þqþ3 ; ð24Þ G 1 i q_ ¼ Cq33 þ X1 X2 ðeiD1 t eiD2 t Þq3 2 G i X1 X2 ðeiD1 t eiD2 t Þq3 ; ð25Þ G where C ¼ r1 þ r2 . We can see from the above equations that, when D1 ¼ D2 ¼ D, equations can become of a very simply form. The explicit timedependent factors of the complex exponential in
q_ 033 ¼ Cq033 þ iGðq03þ q0þ3 Þ; C 0 q_ 3þ ¼ iD q03þ þ iGðq033 q0þþ Þ; 2 C 0 q_ 3 ¼ iD q03 iGq0þ ; 2 C 0 q þ iGðq0þ3 q03þ Þ; 2 33 ¼ iGq03 ;
ð27Þ ð28Þ ð29Þ
q_ 0þþ ¼
ð30Þ
q_ 0þ
ð31Þ
q_ 0 ¼
C 0 q : 2 33
ð32Þ
Eq. (32) has only a decay term with decay rate C=2; it shows the population to be trapped in the ji, while the population is cycled between the levels j3i and jþi due to the q effective field ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipumping ffi
with Rabi frequency G ¼ X21 þ X22 Hence, the condition D1 ¼ D2 is of very importance to the generation of the CPT and EIT effects. In the case of the three-level system driven by a single field, as shown in Fig. 1(b), we know that two detunings of the applied field are correlative, e.g. D2 ¼ D1 x21 , which means that the condition D1 ¼ D2 cannot be satisfied, therefore, the CPT state cannot be generated in this system. On the other hand, in the four-level system when either D3 ¼ D1 or D3 ¼ D2 , the CPT (double-dark states) and EIT can be yielded. Hence, there are two CPT and EIT holes with the scanning of the frequency xL2 or xL1 of the coherent fields. Fig. 6 shows the dynamic evolution of the CPT of K-system in a field-dependent basis. In this case, we can see the CPT developed processes under the initial condition: all populations stay in the jþi, ji states. The population of the superposition states reaches steady
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Y.-F. Li et al. / Optics Communications 202 (2002) 97–102
Fig. 6. Dynamic evolution of the system to the CPT under certain conditions. The solid curve corresponding with the parameters: G ¼ 2, D ¼ 0; C ¼ 2 and dashed curve: G ¼ 5, D ¼ 0; C ¼ 2.
state rapidly and the magnitude of effective pumping field affects the just cycled frequency between the levels j3i and jþi. In conclusion, we show that there are doubledark resonances and EIT effects in our four-level system driven by two coherent fields. The dynamics of these effects are explained in terms of superposition state structuring processes. In the theoretical analysis, we find that the equality of detunings of the two coherent driven fields is of importance to the generation of the double-dark resonances and double-EIT.
References [1] S.E. Harris, J.E. Field, A. Imam oglu, Phys. Rev. Lett. 64 (1990) 1107.
[2] K.-L. Boller, A. Imam oglu, S.E. Harris, Phys. Rev. Lett. 66 (1991) 2593. [3] J.E. Field, K.H. Hahn, S.E. Harris, Phys. Rev. Lett. 67 (1991) 3062. [4] M. Xiao, Yong-qing Li, Shao-Zheng Jin, J. Gea-Banacloche, Phys. Rev. Lett. 74 (1995) 666. [5] D.J. Fulton, S. Shepherd, R.R. Moseley, B.D. Sinclair, M.H. Dunn, Phys. Rev. A 52 (1995) 2302. [6] O. Kocharovskaya, Ya.I. Khanin, Zh. ESKP. Teor. Fiz 90 (1986) 1610. [7] S.E. Harris, Phys. Rev. Lett. 62 (1989) 1033. [8] M.O. Scully, S.-Y. Zhu, A. Gavrielides, Phys. Rev. Lett. 62 (1989) 2813. [9] J.Y. Gao et al., Opt. Commun. 93 (1992) 323. [10] H.M. Doss, L.M. Narducci, M.O. Scully, J.Y. Gao, Opt. Commun. 95 (1993) 57. [11] M.O. Scully, Phys. Rev. Lett. 67 (1991) 1855. [12] M. Fleischhauer, C.H. Keitel, M.O. Scully, Chang Su, B.T. Ulrich, Shi-Yao Zhu, Phys. Rev. A 46 (1992) 1468. [13] A.S. Zibrov et al., Phys. Rev. Lett. 76 (1996) 3935. [14] (a) S. Schiemann, A. Kuhn, S. Steuerwald, K. Bergmann, Phys. Rev. Lett. 71 (1993) 3637; (b) J. Oreg, F.T. Hioe, J.H. Eberly, Phys. Rev. A 29 (1984) 690. [15] Peng Zhou, S. Swain, Phys. Rev. Lett. 77 (1996) 3995. [16] Peng Zhou, S. Swain, Phys. Rev. Lett. 78 (1997) 832. [17] T. Pellizzari, S.A. Gardiner, J.I. Cirac, P. Zoller, Phys. Rev. Lett. 75 (1995) 3788. [18] M. SargentIII, M.O. Scully, W.E. Lamb Jr., Laser Physics, Addision-Wesley, Reading, MA, 1974 (Chapter 2, p. 18; Chapter 16, p. 273). [19] Peng Zhou, S. Swain, Phys. Rev. A 56 (1997) 301. [20] M.D. Lukin, S.F. Yelin, M. Fleischhauer, M.O. Scully, Phys. Rev. A 60 (1999) 3225.
Optics Communications 202 (2002) 97–102 www.elsevier.com/locate/optcom
Laser-induced double-dark resonances and double-transparencies in a four-level system Yong-Fang Li a,b,*, Jian-Feng Sun a, Xian-Yang Zhang a, Yong-Chang Wang b b
a College of Physics and Information Technology, Shaanxi Normal University, Xi’an 710062, People’s Republic of China Department of Applied Physics, Institute of Modern Physics, Xi’an Jiaotong University, Xi’an 710049, People’s Republic of China
Received 30 August 2001; received in revised form 12 November 2001; accepted 5 December 2001
Abstract We study the generations of dark resonance and electomagnetically induced transparency (EIT) effects in the fourlevel system with two coherent fields coupling the transition from upper level to a pair of ground levels and a lower excited state. The quantum interference between two excited pathways results in double-dark resonances and doubletransparency holes with adjustable frequencies driven into the absorption spectrum in the atomic system. A clear analytical explanation for our numerical results is presented. It is shown that the relation of detunings of two coherent fields plays an important role in the generations of the coherent population trapping and EIT effects. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 42.50.Hz; 42.65.)k
If an atom is prepared in a coherent superposition state, it is possible to cancel absorption or emission under certain condition. This interesting phenomenon is called coherent population trapping (CPT) or dark resonance; it is by now a wellknown concept in optics and laser spectroscopy. Based on this effect, there have been many applications including electromagnetically induced transparency (EIT) [1–5], lasing without inversion [6–10] the resonant enhancement of the refractive index [11–13] and adiabatic population transfer [14]. These applications have been discussed for
*
Corresponding author. E-mail address: yfl[email protected] (Y.-F. Li).
conventional use in the system with either K or N level configuration. In the V-system, Zhou Peng and Swain have performed many works on the ultrasharp lines and the role of quantum interference in probe absorption [15,16]. In the three-level K-system, however, a single dark state was generated. With the growth of experimental and theoretical studies, many new and more complex nonlinear behaviors associated with multilevel atoms interacting with multifrequency laser fields have been considered. In this paper we present a four-level system driven by two coherent fields and discuss in detail the process of the populations of each level and the population transfer between levels. We find double-dark resonances and double-EIT
0030-4018/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 2 ) 0 1 0 7 2 - 6
98
Y.-F. Li et al. / Optics Communications 202 (2002) 97–102
with adjustable frequencies driven into the absorption spectrum that could not occur in the three-level system. We investigate various features of the double-dark resonances and doubleEIT. In comparison with the three-level K-system, we find that the equality of detunings of the two coherent driven fields is of importance to the generation of the double-dark resonances and double-EIT. The double-dark resonances have many interesting applications such as unipolar and bipolar quantum well lasers and the possibility of using double-dark states in adiabatic passage techniques in the quantum computation is also intriguing [17]. The model under consideration has four levels, as displayed in Fig. 1. The transitions from the upper level j3i with energy hx3 to a pair of ground levels j1i with energy hx1 and j2i with energy hx2 and the lower excited j4i with energy hx4 are driven by two coherent fields with frequencies xL1 , xL2 , respectively. These transitions are also coupled by the vacuum modes. If the separations of j3i from j1i and j2i are much larger than that from j4i or vice versa, we can consider that the vacuum mode coupling j3i to j1i and j2i is totally different from that coupling j3i to j4i. Meanwhile, we assume that the vacuum model which couples j3i to j1i and j2i is different. We also assume that the transitions between j4i and j2i, j1i are dipole forbidden. In the absence of coherent field with frequency xL2 the system becomes a three-level system where the upper transition level is coupled by a single driven field with frequency xL displayed in Fig. 1(b). In the absence of splitting of two ground
(a)
(b)
(c)
Fig. 1. Schematic diagrams of the energy levels: (a) four-level system driven by two coherent fields; (b) three-level system driven by a single coherent field; (c) three-level K-system driven by two coherent fields.
levels, the system becomes a typical K-system, as shown in Fig. 1(c). We investigate double-dark states (CPT states) and double-EIT effects corresponding to various level-systems and present a clear analytical explanation for the numerical results. The Hamiltonian in the frame rotating with the laser frequencies xL1 , xL2 is in the form H^ ¼ hD1 A11 hD2 A22 hD3 A44 þ ½X1 A31 þ X1 A32 þ X1 A34 þ H :c:;
ð1Þ
where D1;2 ¼ x3 x1;2 xL1 , D3 ¼ x3 x4 xL2 , are the detunings. Xi ði ¼ 1; 2; 3Þ is the Rabi frequency. They are defined as *
X1;2 ¼ h1 E1 l 31;2 e^1 ; X3 ¼ h
1
* E2 l 3;4
ð2Þ
e^2 ;
ð3Þ
* e 1;2
where E1;2 and are the complex amplitudes and * * basis vectors, respectively. Here, l 31;2 ¼ h3j e r j * * 1; 2i and l 34 ¼ h3j e r j4i are matrix elements of the electric dipole moments of atomic transition from j3i j1i, j2i and j3i j4i, which are assumed to be real in our system. The Alk ¼ jlihkj represents a population operator for l ¼ k and dipole transition operator for l 6¼ k. According to the generalized reservoir theory with the Weisskopf–Wigner approximation [18], and the rotating approximation, we can obtain the equations of motion for the reduced atomic density operator reported as in [19]. 1 q_ ¼ i½H ; q þ r1 ð2A13 qA31 A33 q qA33 Þ 2 1 þ r2 ð2A23 qA32 A33 q qA33 Þ 2 1 þ r3 ð2A43 qA34 A33 q qA33 Þ; ð4Þ 2 where rk ðk ¼ 1; 2; 3Þ is a spontaneous decay constant from the excited j3i to lower sublevels jki * 2 (k ¼ 1; 2; 4) and they are given by r12 ¼ jl 31;2 j ðx3 * 3 2 3 x1;2 Þ =3pe0 hc3 , r3 ¼ jl 34 j ðx3 x4 Þ =3pe0 hc3 , respectively. The equations of motion of the reduced density-matrix elements for the atomic variables take the form q_ 11 ¼ r1 q33 þ iX1 ðq13 q31 Þ; q_ 12 ¼ i½ðD1 D2 Þq12 þ X2 q13
ð5Þ X 1 q32 ;
ð6Þ
Y.-F. Li et al. / Optics Communications 202 (2002) 97–102
1 q_ 13 ¼ ðr1 þ r2 þ r3 Þq13 þ iX1 ðq11 q33 Þ 2 þ iðX2 q12 þ D1 q13 þ X3 q14 Þ;
ð7Þ
q_ 14 ¼ i½X3 q13 þ ðD1 D3 Þq14 X1 q34 ;
ð8Þ
q_ 22 ¼ r2 q33 þ iX2 ðq23 q32 Þ;
ð9Þ
1 q_ 23 ¼ ðr1 þ r2 þ r3 Þq23 2 þ iðX1 q21 þ D2 q23 þ X2 q24 Þ þ X2 ðq22 q33 Þ; ð10Þ q_ 24 ¼ i½X3 q23 ðD2 D3 Þq14 X2 q34 ;
ð11Þ
q_ 33 ¼ ðr1 þ r2 þ r3 Þq33 þ i½X1 ðq31 q13 Þ þ X2 ðq32 q23 Þ þ X3 ðq34 q43 Þ;
ð12Þ
1 q_ 34 ¼ ðr1 þ r2 þ r3 Þq34 iX3 ðq44 q33 Þ 2 iðX1 q14 þ X2 q24 Þ iD3 q34 ;
ð13Þ
q11 þ q22 þ q33 þ q44 ¼ 1:
ð14Þ
Eqs. (5)–(14) can be rewritten in the compact vector form as d ^ ^ þ I^; W ¼ L^W ð15Þ dt ^ is a column vector whose ith component where W is Wi ^5 ¼ W ^ ; W ^ 3 ¼ q13 ; W 1 ^ 13 ¼ W ^ ; W ^ 6 ¼ q22 ; W4 ¼ q14 ; W 4 ^ 10 ¼ W ^ ; W ^ 8 ¼ q24 ; W ^ 14 ¼ W ^ ; W 7 8 ^ 11 ¼ q33 ; W ^ 12 ¼ q34 ; W ^ 15 ¼ W : W ð16Þ 12
^ 1 ¼ q11 ; W ^ 9 ¼ W ; W 3 ^ 7 ¼ q23 ; W
^ 2 ¼ q12 ; W
99
j4i can be obtained by plotting the Im½q34 ð1Þ and Re½q34 ð1Þ, respectively, as a function of the variable detuning D3 . For simplicity, it is assumed that the spontaneous rates are the same for all the levels, r1 ¼ r2 ¼ r3 ¼ r. The Rabi frequencies and detunings Dij are normalized by the spontaneous rate r. First, we present the results for the case which is a three-level system with a single driving field, as shown in Fig. 1(b). Fig. 2(a) exhibits the population distribution, absorption and dispersion spectra. When the driving field is resonantly with transition either j3i j1i or j3i j2i, corresponding to D1 ¼ 0 or D1 ¼ x21 , the populations of the j1i or j2i are minimum. Only when D1 ¼ x21 =2, the population of j3i and absorption spectrum of j3i j1i ðj3i j2iÞ have a peak and the CPT state is absent. In the three-level K-system, however, we can clearly see that the population of upper level j3i is zero and the populations of the j1i and j2i are not minimum, moreover, the former has a slight increase when both driving fields are resonantly with transitions j3i j1i and j3i j2i, respectively, as shown in Fig. 2(b). For this case the populations of level j1i and j2i are about 50%, respectively. It shows that the populations are trapped in the superposition state of j1i and j2i. From the Fig. 2(b) we can see that the absorption spectrum exhibits a transparency hole. In addition, we find that as field intensities increase
L is a 15 15 matrix, and the inhomogeneous term I^ is the 15-element column vector. Hence Eq. (15) has formal solution ^ ðt þ sÞ ¼ esL^W ^ ðtÞ þ L^1 ½esL^ 1I^ W
ð17Þ
and steady-state solution is ^ ð1Þ ¼ L^1 I^: W
ð18Þ
We can numerically calculate population distribution of each level, absorption and dispersion spectra by using the steady-state solution (18). The absorption and dispersion spectra between j3i and
Fig. 2. Calculated population distribution and absorption, dispersion spectrum: (a) the three-level system with a single driving field under conditions: r1 ¼ r2 ¼ 1; X1;2 ¼ 1; x21 ¼ 2; (b) the K-system with two driving fields under conditions: r1 ¼ r2 ¼ 1; X1;2 ¼ 1; D2 ¼ 0.
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the position of the transparency hole remains unchanged, but the hole becomes wider. For the four-level system, the population feature of upper level j3i is greatly different from that of a three-level K-system; it has double-dark resonances or CPT states as shown in Fig. 3. When driving field with frequency xL2 is resonantly with transition j3i j4i and driving field with frequency xL1 is far from resonant frequency, the populations of atomic system assemble mainly at the j1i and j2i, the population of j4i is empty. But when driving field with frequency xL1 is also resonant with transition between j3i and j1i which correspond with D1 ¼ 0, the populations of j1i and j4i increase, while the population of j2i vanishes. When driving field xL1 is resonant with transition between j3i and j2i which correspond with D1 ¼ x21 , the populations of j2i and j4i increase, while the population of j1i vanishes, as shown in Fig. 3 ðq11 ; q22 ; q44 Þ. From the change of the population of each level, we can see the population transfer from the j1i; j2i to j4i in the processes of forming CPT states. In such a case, the generation of the double-dark states is similar to a double three-level K-system [20]. Fig. 4 shows that calculated population of j3i and absorption, dispersion spectral profiles are observed with the j3i j4i transition for the variable detuning of the driving field D3 . For
the case of the driving field xL2 is resonantly with the j3i j4i transition, the positions of the CPT states and transparent holes correspond just to the detuning D3 ¼ D1 or D2 , as shown in Fig. 4. The CPT states and transparence holes can be obtained with the great range of Rabi frequencies shown in Fig. 5. The Rabi frequency X3 only affects the magnitude of the absorption spectra, but does not affect the positions of the transparent holes. Based on [20] and above numerical results, we know that the processes of population trapping in our four-level system are similar to double threelevel K-system. Hence, for understanding the origination of these numerical results, we analyze the density matrix equations of the three-level Ksystem in a field-dependent basis.
Fig. 3. The population distribution for the variable detuning D1 in the four-level system under conditions: r1 ¼ r2 ¼ r3 ¼ 1; X1;2;3 ¼ 1; D3 ¼ 0; x21 ¼ 2.
Fig. 4. Calculated population and absorption, dispersion spectrum versus variable detuning D3 under the conditions: r1 ¼ r2 ¼ r3 ¼ 1; X1;2;3 ¼ 1; D1 ¼ 1; x21 ¼ 2.
j3i; jþi ¼
X1 j1i þ X2 j2i X1 j2i X2 j1i ; ji ¼ ; G G ð19Þ
where the X1;2 is the Rabi frequency of two driving fields and assuming X1;2 is real. The G is the effective Rabi ffi frequency of the pumping field qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 G ¼ X1 þ X22 . The j3i is the upper level and j1i and j2i are ground levels, as shown in Fig. 1(c). In this basis, the density matrix equations of motion can be given
Y.-F. Li et al. / Optics Communications 202 (2002) 97–102
101
Eqs. (20)–(25) can be removed by the transformations q03þ ¼ q3þ eiDt ; q03 ¼ q3 eiDt ;
q03 ¼ q3 eiDt ; q0þ3 ¼ qþ3 eiDt ;
q0 ¼ q ;
q033 ¼ q33 ;
q0þ ¼ qþ ;
q0þ ¼ qþ :
q0þþ ¼ qþþ ; ð26Þ
Substituting Eq. (26) into Eqs. (20)–(25), then the equations can be rewritten in the form Fig. 5. Calculated absorption spectra versus variable detuning D3 and Rabi frequency X3 under conditions: r1 ¼ r2 ¼ r3 ¼ 1; X1;2 ¼ 1; D1 ¼ 1; x21 ¼ 2.
i ½X1 X2 ðeiD1 t eiD2 t Þðq3 q3 Þ G ð20Þ þ ðX21 eiD1 t þ X22 eiD2 t Þðq3þ qþ3 Þ;
q_ 33 ¼ Cq33 þ
1 i q_ 3þ ¼ Cq3þ þ ½X1 X2 ðeiD1 t eiD2 t Þqþ 2 G þ q33 ðX21 eiD1 t þ X22 eiD2 t Þ i qþþ ðX21 eiD1 t þ X22 eiD2 t Þ; ð21Þ G 1 i q_ 3 ¼ r Cq3 þ X1 X2 ðeiD1 t eiD2 t Þðq q33 Þ 2 G i 2 iD1 t q3 ðX1 e þ X22 eiD2 t Þ; ð22Þ G 1 i q_ þþ ¼ Cq33 þ ðX21 eiD1 t þ X22 eiD2 t Þqþ3 2 G i 2 iD1 t ðX1 e þ X22 eiD2 t Þq3þ ; ð23Þ G i q_ þ ¼ ðX21 eiD1 t þ X22 eiD2 t Þq3 G i X1 X2 ðeiD1 t eiD2 t Þqþ3 ; ð24Þ G 1 i q_ ¼ Cq33 þ X1 X2 ðeiD1 t eiD2 t Þq3 2 G i X1 X2 ðeiD1 t eiD2 t Þq3 ; ð25Þ G where C ¼ r1 þ r2 . We can see from the above equations that, when D1 ¼ D2 ¼ D, equations can become of a very simply form. The explicit timedependent factors of the complex exponential in
q_ 033 ¼ Cq033 þ iGðq03þ q0þ3 Þ; C 0 q_ 3þ ¼ iD q03þ þ iGðq033 q0þþ Þ; 2 C 0 q_ 3 ¼ iD q03 iGq0þ ; 2 C 0 q þ iGðq0þ3 q03þ Þ; 2 33 ¼ iGq03 ;
ð27Þ ð28Þ ð29Þ
q_ 0þþ ¼
ð30Þ
q_ 0þ
ð31Þ
q_ 0 ¼
C 0 q : 2 33
ð32Þ
Eq. (32) has only a decay term with decay rate C=2; it shows the population to be trapped in the ji, while the population is cycled between the levels j3i and jþi due to the q effective field ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipumping ffi
with Rabi frequency G ¼ X21 þ X22 Hence, the condition D1 ¼ D2 is of very importance to the generation of the CPT and EIT effects. In the case of the three-level system driven by a single field, as shown in Fig. 1(b), we know that two detunings of the applied field are correlative, e.g. D2 ¼ D1 x21 , which means that the condition D1 ¼ D2 cannot be satisfied, therefore, the CPT state cannot be generated in this system. On the other hand, in the four-level system when either D3 ¼ D1 or D3 ¼ D2 , the CPT (double-dark states) and EIT can be yielded. Hence, there are two CPT and EIT holes with the scanning of the frequency xL2 or xL1 of the coherent fields. Fig. 6 shows the dynamic evolution of the CPT of K-system in a field-dependent basis. In this case, we can see the CPT developed processes under the initial condition: all populations stay in the jþi, ji states. The population of the superposition states reaches steady
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Fig. 6. Dynamic evolution of the system to the CPT under certain conditions. The solid curve corresponding with the parameters: G ¼ 2, D ¼ 0; C ¼ 2 and dashed curve: G ¼ 5, D ¼ 0; C ¼ 2.
state rapidly and the magnitude of effective pumping field affects the just cycled frequency between the levels j3i and jþi. In conclusion, we show that there are doubledark resonances and EIT effects in our four-level system driven by two coherent fields. The dynamics of these effects are explained in terms of superposition state structuring processes. In the theoretical analysis, we find that the equality of detunings of the two coherent driven fields is of importance to the generation of the double-dark resonances and double-EIT.
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